Ergodic Capacity and Symbol Error Rate Analysis of a Wireless System with α-μ Composite Fading Channel

A composite α-µ/Lognormal fading channel is proposed with several channel performance criteria. This model considers the most effective occurrences in a fading channel, mainly non-linearity, multi-cluster nature of propagation medium, and shadowing effects. The new generation of communication systems is moving towards the use of millimetre waves (mmW). In this type of propagation, large-scale effects of fading channel on the received signal are significant, so in the proposed composite model, the lognormal distribution is considered to model large-scale effects of fading, which is the most accurate distribution to model shadowing. The Gaussian-Hermite quadrature sum is used to approximate the probability distribution function (PDF) of the proposed model. After calculating the statistics, the symbol error rate (SER) and ergodic capacity are computed. The Mellin transform technique is used to calculate the SER expression of different modulation schemes; then, ergodic capacity is computed for a diverse frequency spectrum. Finally, the Monte Carlo method is used to evaluate the analyses.


Introduction
Fading is one of the most destructive effects observed over wireless communication channels on a propagated signal. It is typically described by two general phenomena ' multi-path and shadowing'. Shadowing is the slow attenuation of signal known as largescale fading or long-term fading. It is called a large-scale fading since it can be observed at several tens of wavelengths. According to some experiments on the indoor and outdoor field capacity, lognormal distribution well-models the shadowing [1]. Multi-path is also caused by scatterers, such as the foliage, buildings, and the obstacles between transmitter and receiver. It makes rapid changes in the received signal amplitude and occurs at a fraction of the wavelength; hence, it is called small-scale fading. So far, • Computation of SER for the proposed model concerning coherent BFSK, M-ary ASK, M-ary PSK, as well as M-ary QAM and differential BPSK modulation schemes. • Calculation of ergodic capacity for the proposed composite channel.

α-µ Distribution
Let the variable, U indicates the received signal amplitude and contains α-μ distribution, then its probability density function can be represented as [4,Eq. 1], where, > 0 is an arbitrary fading parameter implying non-linearity, > 0 represents multi-path clusters, û =

Lognormal Distribution
As previously mentioned, the Lognormal is a distribution to model the shadowing phenomenon in the channel. The probability density function of the received signal amplitude ω, can be expressed as follows [1] where σ and m are the standard deviation and mean of the lognormal distribution, respectively.

Composite Fading
Let Z be the received signal amplitude with α-µ/lognormal composite distribution; then its PDF is computed by averaging the conditional probability density function of α-μ over lognormal distribution as follows Also, the conditional PDF of received signal amplitude can be expressed as follows . By placing (4) and (2) into (3), we achieve The integral in (5) does not lead to the closed-form expression, so it is rewritten in an approximate solvable form. By substitution, x = log e ( ) √ 2 , we get where, The integral in (6)

Cumulative Distribution Function (CDF)
The CDF can be calculated using F Z (z) = ∫ z 0 f Z (t)dt that yields, According to Appendix A, the above expression can be displayed in the form below, where, (. ; .) is the lower incomplete gamma function defined as (s;x) = ∫ x 0 t s−1 e −t dt.

SNR Distribution Approximation
The variable Z is considered as the received signal amplitude, so the instantaneous SNR at the receiver can be expressed as [22], where, ̃= E[ẑ 2 ]E s N 0 and E s N 0 is the energy per bit to the noise power spectral density ratio. The SNR PDF is calculated as follows, The cumulative distribution function of instantaneous SNR is also obtained as, The kth moment of γ can be expressed as, (14), we get The integral in the above relation is in the form of the gamma function, so

Amount of Fading (AF)
The amount of fading (AF) is a statistical property representing the severity of fading and according to [1], the AF can be calculated as follow, where, the random variable γ is the received signal SNR, and the above expression can be calculated using the second and fourth moments; hence, the AF equation can be expressed as,

Outage Probability (Pout)
The outage probability is one of the main benchmarks of channel operation. The outage probability is the probability that the error rate being beyond a predetermined threshold or, in other words, the probability that the received SNR being below a given threshold, th so it can be easily calculated as, where,F ( ) is the cumulative distribution function.

Expressions for Symbol Error Rate
In this section, the symbol error probability for different coherent and non-coherent modulation schemes is calculated. The method based on the Mellin transform is employed to compute the SER, which is introduced in [17]. This transform is defined as follows [23]: The inverse of the Mellin transform is also defined as Now, if we assume that f ( ) and F( ) are the instantaneous SNR probability density function (PDF) and channel cumulative distribution function (CDF), respectively, then SER can be calculated as follows [17,Eq , and c is considered as a constant, which is defined in the strip of f * (s) . The expression P(error| ) is also considered as the probability of a conditional error, which is dependent on the employed modulation technique. In order to calculate the error probability of the proposed channel model, primarily, the Mellin transform of the channel CDF is calculated.
Appendix B represents the procedure in which the above expression is calculated. By placing (23) in (22), SER can be expressed as As can be seen from Table 1, the expression (−d∕d P(error| )) is taken into account as a linear combination of factors [17]. The expression Q � (.) is related to the complement of the Gaussian-Q function.
In (22), g * (s) is a linear combination of the following terms: (13)]. The unconditional SER of these expressions is called I 0 (b),I 1 (b) , and I 2 (a, b) , respectively. Table 2 shows the value of former expressions based on (24), in which H m,b p,q (z) is defined as the Fox-H function and H 0,n;m 1 ,n 1 ;...;m L ,n L p,q;p 1 ,q 1 ;...p l ,q l (z 1 , ..., z L ) is described as the multivariate Fox-H function [24]. Now, by substituting the attained expressions for I 0 (b) , I 1 (b) and I 2 (a, b) regarding [17 Table 2], the SER is calculated for different modulations. The error probability is given for various modulation schemes in Table 3.

Asymptotic Analysis
In practical applications, for simplicity of calculations and non-use of Fox-H function expressions, it is tended to calculate asymptotic expressions for high SNRs. Theories [24, Theorem 1.2, and Theorem 1.3] can be used to calculate asymptotic expressions of Fox-H function. Based on [17], the complex residue theorem is used to calculate the asymptotic expressions for I 0 (b),I 1 (b) , and I 2 (a, b) , so that only the pole closest to the contour is considered. Also, for large SNRs, only the poles lying to the right of the contour are   The asymptotic expression for large SNRs is then calculated as follows

Ergodic Capacity
The ergodic capacity is the maximum available rate considering the fading channel, which is defined as follows where f ( ) is the channel probability distribution function, γ is the instantaneous SNR and W is the bandwidth of the channel. Substituting (12) in (29)  where, Δ(n, a) = a n , a+1 n , ..., a+n−1 n , in which a is a real value, n is a positive integer, k and k 2 are positive integer, such that gcd(L, k) = 1 and G m,n p,q is the Meijer's G-function [26].

Numerical Results
In this section, the numerical and analytical results of the expressions computed in the previous sections are presented. In all the introduced equations, the n value for Gaussian Hermite polynomial is set as 20. In order to generate a random variable with the proposed distribution, primarily random variables with an α-μ distribution are generated according (28) � to [27]. Then, by multiplying these variables in random variables generated by a lognormal distribution, random variables by the proposed distribution are obtained. For evaluating the accuracy of the proposed distribution, the exact and approximated PDF graphs are plotted in Fig. 1. As can be observed, there is a close match between the exact and approximate figures. By setting = 2 , the graph shows the composite Nakagami-m/Lognormal distribution, and Rayleigh/Lognormal distribution is obtained by placing α = 1.
In Fig. 2  α-μ / Lognormal fading model covers more channel conditions than many other channel models, which shows the importance of this distribution.
In Figs. 3 and 4, SER graphs are plotted for CBFSK, QPSK, DBPSK, M-ary ASK, and M-ary QAM digital modulation schemes. In these figures, solid lines represent the approximated expressions while dot ones represent the Monte Carlo simulations, and the dashed line shows the asymptotic graphs. It is obvious, there is a good match between accurate and simulated diagrams. Asymptotic diagrams in large SNRs also match to the graphs. Figure 3 shows that binary modulation has the lowest error probability compared to other modulations. In Fig. 4, the SER is plotted for the eight and 16-ary modulation schemes. As expected, in all 16-ary systems, the probability of error is higher than 8-ary ones. Figure 5 shows the SER of QPSK modulation under different composite models, which can be obtained from the α-µ/Lognormal. The negative exponential/lognormal distribution can be achieved by setting α = 1 and μ = 1 in α-µ/Lognormal distribution, and one-sided Gaussian/lognormal distribution is obtained by setting α = 2 and μ = 0.5. The Nakagami-m model is also a special case of the proposed model, which is obtained by placing α = 2 in the α-μ / Lognormal relation, and the 'µ' parameter is equivalent to the 'm' parameter in the Nakagami-m/Lognormal model. By setting µ = 1 in the α-μ / Lognormal model, the distribution of Weibull/Lognormal is obtained, in which case 'K' in the distribution of Weibull/Lognormal will be equivalent to 'α' in the distribution of the α-μ / Lognormal. As can be seen, the probability of error in Rayleigh/lognormal, one-sided Gaussian/lognormal, and negative exponential/lognormal models is higher than models such as Weibull/ lognormal or Nakagami-m/lognormal, because in Nakagami-m/lognormal, the multi-cluster nature of the environment and in Weibull/lognormal the non-linearity of the propagation medium is considered.  and analytical graphs. As can be expected, the capacity has enhanced by increasing bandwidth and reached about a few Gbps in the millimeter-wave spectrum.

Concusions
The channel algebraic representations are vital in digital communication studies. In this paper, the expressions for the PDF, CDF, and the moments of the proposed composite α-µ/Lognormal channel model concerning both the signal amplitude and the instantaneous SNR are presented. Symbol error rates are also calculated for different modulation techniques, including CBFSK, QASK, QPSK, 16-QAM, and DBPSK, by using Mellin transform method. The ergodic capacity of the proposed channel is also calculated in the form of the Meijer-G function. The positive features of this distribution are its comprehensiveness and considering the critical factors influencing the fading, such as multipath, multi-cluster, non-linearity, and shadowing. This distribution can be used to model millimetre-wave propagation channels.

Appendix A
To calculate the CDF, according to the definition, F Z (z) = ∫ The integral is in the form of an incomplete gamma function, so